Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = quantity x minus seven divided by quantity x plus two. and g(x) = quantity negative two x minus seven divided by quantity x minus one.

F(x) = (x - 7)/(x + 2)
g(x) = (-2x - 7)/(x - 1)
f(g(x)) = [(-2x - 7)/(x-1) - 7]/[(-2x - 7)/(x-1) + 2]
= [(-2x - 7 - 7x + 7)/(x - 1)]/[(-2x - 7 + 2x - 2)/(x - 1)]
=(-9x)/-9
= x

g(x) = (-2x - 7)/(x - 1)
g(f(x)) = [-2(x - 7)/(x + 2) - 7]/[(x - 7)/(x + 2) - 1]
= [(-2x + 14 - 7x - 14)/(x + 2)]/[(x - 7 - x - 2)/(x + 2)]
= -9x/-9
= x

i.e. f(g(x)) = g(f(x)) = x
which shows that they are inverses.